Fundamentals of Aerodynamics 2024, 46: Inviscid, Compressible Flow

Inviscid, compressible, subsonic flow over a body immersed in a uniform stream is irrotational, so a velocity potential for such flows can always be defined. Such potential requires continuity, momentum and energy equations to work together correctly, and hence may be derived from those. With the use of isentropy of the potential, and mostly straight-forward substitution, gain a differential equation for the potential ϕ

The potential then is derived from a nonlinear PDE, which are notoriously difficult to solve analytically. Often, solutions to these will be approximated numerically. Perturbations in the flow of this potential can be narrowed down to perpendicular parts of the motion lines. These also get their own velocity potential. It's written largely the same, though the infinite velocity potential of course figures into the coordinate component chosen to align with it. Due to these perturbations being assumed to be small, in subsonic Mach regimes, they can be ignored. In supersonic regimes,

Compressibility corrections transitions the latter quantity into the Prandtl-Glauert rule, relating the incompressible and compressible pressure distributions over an airfoil. This in turn should give the lift and momentum coefficients.

This can be further improved through either Karman-Tsien or Laitone's rule, though these are basically second- and third order perturbation. At critical Mach-number, the pressure coefficient changes.

An airfoil at a fixed angle of attack in a wind tunnel will develop its drag coefficient with the Mach number. It begins rising rapidly after the critical Mach number and fall off after reaching the sound barrier. Said critical Mach number is also referred to as the "drag-divergence Mach number". It's naturally typically below 1.